AOE 5104 Class 10

1. AOE 5104 Class 10 Online presentations for next class: Potential Flow 1 Homework 4 due 10/2

2. Kinematics of Vorticity

3. Kinematic Concepts - Vorticity

4. Spinning Cone

5. Implications (Helmholtz� Vortex Theorems, Part 1) The strength of a vortex tube (defined as the circulation around it) is constant along the tube. The tube, and the vortex lines from which it is composed, can therefore never end. They must extend to infinity or form loops. The average vorticity magnitude inside a vortex tube is inversely proportional to the cross-sectional area of the tube Rotating Drum Bathtub vortexRotating Drum Bathtub vortex

6. Vortex Tube

7. Implications (Helmholtz� Vortex Theorems, Part 1) The strength of a vortex tube (defined as the circulation around it) is constant along the tube. The tube, and the vortex lines from which it is composed, can therefore never end. They must extend to infinity or form loops. The average vorticity magnitude inside a vortex tube is inversely proportional to the cross-sectional area of the tube Rotating Drum Bathtub vortexRotating Drum Bathtub vortex

9. Kinematic Concepts - Vorticity

10. But, does the vortex tube travel along with the fluid, or does it have a life of it�s own?

12. Body Force Torque

13. Pressure Force Torque http://www3.imperial.ac.uk/earthscienceandengineering/research/computationalphysics/oceancirculation/freesurface/sloshingtank/http://www3.imperial.ac.uk/earthscienceandengineering/research/computationalphysics/oceancirculation/freesurface/sloshingtank/

14. Shock in a CD Nozzle M1=1.81, rho2/rho1=2.38, p2/p1=3.66, M2=0.61, U1~628m/s (if T=300K). If b.l. 1mm thick, and shear accomodated by simple rotation of the fluid particles then rotation rate = 200000 times per secondM1=1.81, rho2/rho1=2.38, p2/p1=3.66, M2=0.61, U1~628m/s (if T=300K). If b.l. 1mm thick, and shear accomodated by simple rotation of the fluid particles then rotation rate = 200000 times per second

15. Viscous Force Torque Viscous force torques are non-zero where viscous forces are present ( e.g. Boundary layer, wakes) Can be really small, even in viscous regions at high Reynolds numbers since viscous force is small in that case The viscous force torques can then often be ignored over short time periods or distances

16. Shock in a CD Nozzle M1=1.81, rho2/rho1=2.38, p2/p1=3.66, M2=0.61, U1~628m/s (if T=300K). If b.l. 1mm thick, and shear accomodated by simple rotation of the fluid particles then rotation rate = 200000 times per secondM1=1.81, rho2/rho1=2.38, p2/p1=3.66, M2=0.61, U1~628m/s (if T=300K). If b.l. 1mm thick, and shear accomodated by simple rotation of the fluid particles then rotation rate = 200000 times per second

17. Implications In the absence of body-force torques, pressure torques and viscous torques� the circulation around a fluid loop stays constant: Kelvin�s Circulation Theorem a vortex tube travels with the fluid material (as though it were part of it), or a vortex line will remain coincident with the same fluid line the vorticity convects with the fluid material, and doesn�t diffuse fluid with vorticity will always have it fluid that has no vorticity will never get it

18. Lord Kelvin (1824-1907) (William Thompson ) Lord Kelvin, of course! (William Thompson) Scottish mathematician and physicist who contributed to many branches of physics. He was known for his self-confidence, and as an undergraduate at Cambridge he thought himself the sure "Senior Wrangler" (the name given to the student who scored highest on the Cambridge mathematical Tripos exam). After taking the exam he asked his servant, "Oh, just run down to the Senate House, will you, and see who is Second Wrangler." The servant returned and informed him, "You, sir!" (Campbell and Higgens, p.�98, 1984). Another example of his hubris is provided by his 1895 statement "heavier-than-air flying machines are impossible" (Australian Institute of Physics), followed by his 1896 statement, "I have not the smallest molecule of faith in aerial navigation other than ballooning...I would not care to be a member of the Aeronautical Society." Kelvin is also known for an address to an assemblage of physicists at the British Association for the advancement of Science in 1900 in which he stated, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement." A similar statement is attributed to the American physicist Albert Michelson. Kelvin argued that the key issue in the interpretation of the Second Law of Thermodynamics was the explanation of irreversible processes. He noted that if entropy always increased, the universe would eventually reach a state of uniform temperature and maximum entropy from which it would not be possible to extract any work. He called this the Heat Death of the Universe. With Rankine he proposed a thermodynamical theory based on the primacy of the energy concept, on which he believed all physics should be based. He said the two laws of thermodynamics expressed the indestructibility and dissipation of energy. He also tried to demonstrate that the equipartition theorem was invalid. Thomson also calculated the age of the earth from its cooling rate and concluded that it was too short to fit with Lyell's theory of gradual geological change or Charles Darwin's theory of the evolution of animals though natural selection. He used the field concept to explain electromagnetic interactions. He speculated that electromagnetic forces were propagated as linear and rotational strains in an elastic solid, producing "vortex atoms" which generated the field. He proposed that these atoms consisted of tiny knotted strings, and the type of knot determined the type of atom. This led Tait to study the properties of knots. Kelvin's theory said ether behaved like an elastic solid when light waves propagated through it. He equated ether with the cellular structure of minute gyrostats. With Tait, Kelvin published Treatise on Natural Philosophy (1867), which was important for establishing energy within the structure of the theory of mechanics. (It was later republished under the title Principles of Mechanics and Dynamics by Dover Publications). Lord Kelvin, of course! (William Thompson) Scottish mathematician and physicist who contributed to many branches of physics. He was known for his self-confidence, and as an undergraduate at Cambridge he thought himself the sure "Senior Wrangler" (the name given to the student who scored highest on the Cambridge mathematical Tripos exam). After taking the exam he asked his servant, "Oh, just run down to the Senate House, will you, and see who is Second Wrangler." The servant returned and informed him, "You, sir!" (Campbell and Higgens, p.�98, 1984). Another example of his hubris is provided by his 1895 statement "heavier-than-air flying machines are impossible" (Australian Institute of Physics), followed by his 1896 statement, "I have not the smallest molecule of faith in aerial navigation other than ballooning...I would not care to be a member of the Aeronautical Society." Kelvin is also known for an address to an assemblage of physicists at the British Association for the advancement of Science in 1900 in which he stated, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement." A similar statement is attributed to the American physicist Albert Michelson. Kelvin argued that the key issue in the interpretation of the Second Law of Thermodynamics was the explanation of irreversible processes. He noted that if entropy always increased, the universe would eventually reach a state of uniform temperature and maximum entropy from which it would not be possible to extract any work. He called this the Heat Death of the Universe. With Rankine he proposed a thermodynamical theory based on the primacy of the energy concept, on which he believed all physics should be based. He said the two laws of thermodynamics expressed the indestructibility and dissipation of energy. He also tried to demonstrate that the equipartition theorem was invalid. Thomson also calculated the age of the earth from its cooling rate and concluded that it was too short to fit with Lyell's theory of gradual geological change or Charles Darwin's theory of the evolution of animals though natural selection. He used the field concept to explain electromagnetic interactions. He speculated that electromagnetic forces were propagated as linear and rotational strains in an elastic solid, producing "vortex atoms" which generated the field. He proposed that these atoms consisted of tiny knotted strings, and the type of knot determined the type of atom. This led Tait to study the properties of knots. Kelvin's theory said ether behaved like an elastic solid when light waves propagated through it. He equated ether with the cellular structure of minute gyrostats. With Tait, Kelvin published Treatise on Natural Philosophy (1867), which was important for establishing energy within the structure of the theory of mechanics. (It was later republished under the title Principles of Mechanics and Dynamics by Dover Publications).

21. Vorticity Transport Equation The kinematic condition for convection of vortex lines with fluid lines is found as follows

25. Starting Vortex

26. Example: Flow over a depression in a river bed

27. The River Avon nearing flood stage, Salisbury, Wiltshire, UK

28. Example: Evolution of turbulence in a shear flow

29. Eddies in a turbulent channel flow

30. Example: Flow around a corner in a channel

31. Turbulent flow in a pipe through a 180o bend. Hitoshi Sugiyama, Utsunomiya University

32. Ox Bows